Codimension two submanifolds of positive curvature

Abstract

In this note it is proven that a compact connected n-dimensional Riemannian manifold of positive curvature, isometrically immersed in ( n + 2 ) (n + 2) -dimensional Euclidean space, is a homotopy sphere if n ⩾ 3 n \geqslant 3 ; hence it is homeomorphic to a sphere if n ⩾ 5 n \geqslant 5 .